Finite Group Databasebeta

← C2×#20241324 #26125397

Subgroups — C2×#20241324 #26125397

Order 12032.

626 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688905C212yesmaximal, complemented
2#20241324C214yesmaximal, direct product factor
3C2×C47×#407C211yesmaximal, characteristic
4C2×#13093964—22no—
5#15688905—22no—
6#15688905C2214yescomplemented
7C24×C4×C47C411yescharacteristic
8C22×M16×C47C411yescharacteristic
9C22×M16×C47C2211yescharacteristic
10C47×#407—22no—
11C47×#407C412yes—
12C47×#407C2212yes—
13C2×C22⋊C8×C47—22no—
14#13093964—24no—
15#13093964—42no—
16C2×C47⋊C16—42no—
17C25×C47C811yescharacteristic
18C23×C4×C47—21no—
19C23×C4×C47—21no—
20C23×C4×C47—21no—
21C23×C4×C47—21no—
22C23×C4×C47—41no—
23C23×C4×C47—41no—
24C23×C4×C47—41no—
25C23×C4×C47—41no—
26C23×C4×C47C811yescharacteristic
27C23×C4×C47C812yes—
28C23×C4×C47C2×C411yescharacteristic
29C23×C4×C47C2×C412yes—
30C2×M16×C47—21no—
31C2×M16×C47—41no—
32C2×M16×C47—41no—
33C2×M16×C47—41no—
34C2×M16×C47—41no—
35C2×M16×C47C2×C412yes—
36C2×M16×C47C2×C411yescharacteristic
37C2×M16×C47D412yes—
38C2×M16×C47C2311yescharacteristic
39C22×C8×C47—21no—
40C22×C8×C47D412yes—
41C22⋊C8×C47—24no—
42C22⋊C8×C47—24no—
43C24×C47—21no—
44C24×C47—21no—
45C24×C47—41no—
46C24×C47—41no—
47C24×C47—44no—
48C24×C47C2×C811yescharacteristic
49C24×C47C2×C812yes—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47—42no—
68C22×C4×C47—42no—
69C22×C4×C47—42no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C2×C811yescharacteristic
79C22×C4×C47C2×C812yes—
80C22×C4×C47M1611yescharacteristic
81C22×C4×C47M1611yescharacteristic
82C22×C4×C47C22×C411yescharacteristic
83M16×C47—41no—
84M16×C47—42no—
85M16×C47—41no—
86M16×C47—42no—
87M16×C47—42no—
88C2×C8×C47—22no—
89C2×C8×C47—22no—
90C2×C8×C47—22no—
91C2×C8×C47—41no—
92C2×C8×C47—41no—
93C2×C8×C47C22⋊C412yes—
94C2×C8×C47C2×D412yes—
95C47⋊C16—44no—
96C23×C47—24no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—22no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—44no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47—44no—
106C23×C47—44no—
107C23×C47—48no—
108C23×C47—44no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47C22⋊C811yescharacteristic
112C23×C47C22⋊C811yescharacteristic
113C23×C47C22×C811yescharacteristic
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—22no—
125C2×C4×C47—44no—
126C2×C4×C47—44no—
127C2×C4×C47—42no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47C22⋊C811yescharacteristic
136C2×C4×C47C22⋊C811yescharacteristic
137C2×C4×C47#9911yescharacteristic
138C2×C4×C47#9911yescharacteristic
139C2×C4×C47#10011yescharacteristic
140C2×C4×C47#10011yescharacteristic
141C2×C4×C47C2×C22⋊C411yescharacteristic
142C2×C4×C47C2×M1611yescharacteristic
143C8×C47—22no—
144C8×C47—22no—
145C8×C47—42no—
146C2×#1320—471nomaximal, Sylow
147C22×C47—22no—
148C22×C47—24no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—42no—
154C22×C47—44no—
155C22×C47—42no—
156C22×C47—44no—
157C22×C47—44no—
158C22×C47—48no—
159C22×C47—44no—
160C22×C47—42no—
161C22×C47—42no—
162C22×C47#32311yescharacteristic
163C22×C47#32311yescharacteristic
164C22×C47C2×C22⋊C811yescharacteristic
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—42no—
168C4×C47#32311yescharacteristic
169C4×C47#32311yescharacteristic
170C4×C47C2×#9911yescharacteristic
171C4×C47C2×#10011yescharacteristic
172C2×#407D4711yescharacteristic
173C2×#349—472no—
174#1320—474no—
175C2×C47—21no—
176C2×C47—21no—
177C2×C47—44no—
178C2×C47—41no—
179C2×C47—41no—
180C2×C47#132012yescomplemented
181C2×C47C2×#32311yescharacteristic
182C24×C4Dic4711yescharacteristic
183C22×M16Dic4711yescharacteristic
184C22×M16C2×D4711yescharacteristic
185C2×M32—942no—
186#407—22no—
187#407Dic4712yes—
188#407C2×D4712yes—
189C2×C22⋊C8—22no—
190#349—474no—
191#349—942no—
192C47C2×#132011yescharacteristic, complemented, Sylow
193C25C47⋊C811yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4C47⋊C811yescharacteristic
203C23×C4C47⋊C812yes—
204C23×C4C2×Dic4711yescharacteristic
205C23×C4C2×Dic4712yes—
206C2×M16—21no—
207C2×M16—41no—
208C2×M16—41no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16C2×Dic4712yes—
212C2×M16C2×Dic4711yescharacteristic
213C2×M16C22⋊D4712yes—
214C2×M16C22×D4711yescharacteristic
215C22×C8—21no—
216C22×C8C22⋊D4712yes—
217M32—944no—
218M32—1882no—
219C2×C16—1882no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C24—21no—
223C24—21no—
224C24—41no—
225C24—41no—
226C24—44no—
227C24C2×C47⋊C811yescharacteristic
228C24C2×C47⋊C812yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×C47⋊C811yescharacteristic
256C22×C4C2×C47⋊C812yes—
257C22×C4#1062532911yescharacteristic
258C22×C4#1062532911yescharacteristic
259C22×C4C22⋊Dic4711yescharacteristic
260C22×C4C22⋊Dic4711yescharacteristic
261C22×C4C22×Dic4711yescharacteristic
262M16—41no—
263M16—42no—
264M16—41no—
265M16—42no—
266M16—42no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8C22⋊Dic4712yes—
273C2×C8C2×C22⋊D4712yes—
274C16—1884no—
275C23—24no—
276C23—22no—
277C23—21no—
278C23—21no—
279C23—42no—
280C23—42no—
281C23—44no—
282C23—42no—
283C23—42no—
284C23—44no—
285C23—44no—
286C23—48no—
287C23—44no—
288C23—41no—
289C23—41no—
290C23C22⋊(C47⋊C8)11yescharacteristic
291C23C22⋊(C47⋊C8)11yescharacteristic
292C23C22×C47⋊C811yescharacteristic
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4C22⋊(C47⋊C8)11yescharacteristic
315C2×C4C22⋊(C47⋊C8)11yescharacteristic
316C2×C4#1309393911yescharacteristic
317C2×C4#1309393911yescharacteristic
318C2×C4#1309394111yescharacteristic
319C2×C4#1309394111yescharacteristic
320C2×C4C2×C22⋊Dic4711yescharacteristic
321C2×C4C2×#1062532911yescharacteristic
322C8—22no—
323C8—22no—
324C8—42no—
325C22—22no—
326C22—24no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—42no—
332C22—44no—
333C22—42no—
334C22—44no—
335C22—44no—
336C22—48no—
337C22—42no—
338C22—44no—
339C22—42no—
340C22#1568884711yescharacteristic
341C22#1568884711yescharacteristic
342C22C2×C22⋊(C47⋊C8)11yescharacteristic
343C4—21no—
344C4—21no—
345C4—42no—
346C4#1568884711yescharacteristic
347C4#1568884711yescharacteristic
348C4C2×#1309393911yescharacteristic
349C4C2×#1309394111yescharacteristic
350C2—21no—
351C2—21no—
352C2—44no—
353C2—41no—
354C2—41no—
355C2#2024132412yesdirect product factor
356C2C2×#1568884711yescharacteristic
subgroupG/H
centerC2×C4#13093939
commutatorC23×C47C22×C8
FrattiniC2×M16C22×D47
FittingC2×C47×#407C2
socleC22×C47#323
prime radicalC25×C47C8
cosocleC2×M16×C47C23
hypercenterC2×#407D47
hypocenterC47C2×#1320
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×M16 ◁ C2×#407

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#15688905C2semidirect
2. #20241324C2direct (N × H)
6. #15688905C22semidirect
180. C2×C47#1320semidirect
192. C47C2×#1320semidirect
355. C2#20241324direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156889052
#202413242
C2×C47×#4072
C2×#132047